Laser sail-based spacecraft-where a powerful Earth-based laser propels a lightweight outer-space vehicle-have been recently proposed by the Breakthrough Starshot Initiative as a means of reaching relativistic speeds for interstellar space travel. The laser intensity at the sail required for this task is at least 1 GW m(-2) and, at such high intensities, thermal management of the sail becomes a significant challenge even when using materials with low linear absorption coefficients. Silicon is proposed as one leading candidate material for the sail due to its low sub-bandgap absorption and high index of refraction, which allows for low-mass-density designs. However, here it is shown that the temperature-dependent linear absorption of silicon can lead to thermal runaway at temperatures above 400-500 K for even the most optimistic viable assumptions of the material quality. Additionally, above a design-specific threshold laser intensity, nonlinear two-photon absorption triggers thermal runaway regardless of initial temperature. Resonator-based designs, which concentrate the field, exhibit lower threshold intensities than geometries that minimize the electric field such as Bragg reflectors.